Front–Concatenation Shuffles on N: Tail Rigidity, Universal Coverage Laws, and a Deterministic Priority–Queue Framework

We introduce a deterministic two--parameter front--concatenation shuffle on the positive integers, equivalently a priority--queue service discipline in which at step~n the element in position~n is expelled, the next m elements are promoted to the front, and the preceding p elements are demoted behind them. Exhaustivity---whether every integer is eventually expelled---is the combinatorial analogue of queue stability. Our first main result is a parameter--rigidity theorem for the $m=1$ family: for every p≥0, the expelled sequence is Eₙ=2n-1 and precisely the odd integers are expelled, despite the p--dependent reorganization of the internal queue state. An infinite family of combinatorial operators, parameterized by an arbitrary non--negative integer, thus produce identical output. We prove a tail rigidity theorem reducing the infinite dynamics to a finite core of length $n+m-2$ for all $(m,p)$, and for $m=2$ we give complete classifications: $(2,0)$ is exhaustive with Eₙ=n+(-1)ⁿ, and $(2,1)$ is exhaustive via an explicit ``$3x-1$'' parity cascade. The remaining $(2,p)$ cases reduce to a floor--dilation map gᵣ(W)=W+ W/r. We then extend the theory to all m≥ 2 and establish two new universal results.A Universal Arithmetic Progression Theorem: for every $(m,p)$ with m≥ 2 and $T:=m+p$, the expelled sequence contains two deterministic APs of stride~T and common difference~$2(T-1)$, proved via the core recursion by tracking the entry--to--expulsion path of each new label.A Coverage Law: the asymptotic fraction of integers expelled up to the running maximum converges to T/\!(2(T-1)), interpolating between full exhaustivity at $T=2$ and the structural density~$1/2$ of the $m=1$ regime as T→∞. This dichotomy---structural non--exhaustivity (only odd integers expelled) versus dynamical near--non--exhaustivity (all integers eventually expelled, but with growing backlog)---is the paper's central conceptual contribution. Finally, the r--adic avoidance set for gᵣ is a Cantor--type fractal of Hausdorff dimension log(r-1)/log r, non--terminating integers have density~$0$, and stopping times are geometrically distributed under uniform base--r digits.